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MTH6158: Ring Theory

创建日期：2024-08-19 15:25:48

所属分类：数学mathematics

标签： MTH6158

Ring Theory

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MTH6158: Ring Theory

Question 1 [28 marks].

(a) Is the function f : Z → Z defined as f(x) = 3x a ring homomorphism? Justify your answer.

(b) Consider the map ? : Z → Z/10Z defined as ?(m) = [5m]10 .

(i) Prove that ? is a ring homomorphism.

(ii) Describe explicitly the image Im(?) and the kernel Ker(?) of ?. How many elements do Im(?) and Ker(?) have?

(iii) Use the First Isomorphism Theorem to express your answer for Im(?) in part (ii) as isomorphic to a factor ring of Z. Write down explicitly the isomorphism between these two rings.

Question 2 [24 marks]. Let R be the Boolean ring R = P({a,b,c}), with addition being symmetric difference and multiplication being intersection. Consider the ideals I = P({a}) and J = P({b,c}) of R.

(b) Write down explicitly the partition of R into these cosets.

(d) Use the third isomorphism theorem to show that the factor ring R/J is isomorphic to the subring I.

(e) Use the second isomorphism theorem to conclude that if K is an ideal of R containing J then K = J or K = R.

Question 3 [26 marks].

(a) Explain why S is an integral domain.

(b) Is the subset T = {c + 2d√?3 : c,d ∈ Z} an ideal of S? Justify your answer.

(c) Determine all the units of S. Justify your answer.

(d) Provide an example showing that S is not a unique factorisation domain. You do not need to prove anything.

(e) Is the element 7 + √?3 ∈ S irreducible? Justify your answer.

Question 4 [22 marks].

(a) Can the ring R be an integral domain? Explain.

(b) Explain why no element of R can be a unit.

(c) Prove that for any a,b ∈ R we must have a · b = ? (b · a).

(d) Give an example of such a ring R in which there exist a,b ∈ R with a · b 0.

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